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Open a science textbook, and the chances are that somewhere near the
back you will find a list of fundamental constants. This will include things
like the speed of light, c, Newton’s gravitational constant, G; the magnitude
of the charge on an electron, e; and Planck’s constant, h. But are these
numbers fundamental and constant? Or are they mere artefacts of the way
human beings make measurements? By doing away with fundamental constants,
it is possible to paint a simpler picture of the way the Universe works.

A detailed discussion of these ideas inevitably involves mathematics,
but the essence of the argument can be explained in a straightforward, nonmathematical
manner. The best way to begin is by thinking about the speed of light, the
fundamental constant which lies at the heart of Albert Einstein’s special
theory of relativity (¿ìè¶ÌÊÓƵ, Inside Science 49, 21 September 1991).
Imagine sending a pulse of light flashing out into space, and measuring
its distance d at a time t. We write this as d = ct, where c is one of those
fundamental constants that we can look up in a textbook. But where did c
come from?

The time and distance involved are, in a fundamental sense, ‘real’ parameters:
they are directly measurable and they cannot be reduced to anything simpler.
But the constant c was invented, as a way of relating d and t. We need c
because we perceive distance and time to be different concepts. Distance
is a quantity that has the dimensions of length (usually denoted by (L)),
while time is a quantity with dimension (T). So to equate an (L) quantity
with a (T) quantity, we are obliged to introduce a quantity with dimensions
(L)/(T). We call this quantity the speed of light.

EINSTEIN’S CONVERSION FACTOR

The constant c played such an important role in Einstein’s theory because
it allowed him to describe these different concepts of space (distance)
and time as different facets of a single entity: four-dimensional space-time.
As Herman Minkowski realised, space and time can be melded into space-time
if we regard d and ct as the same kind of thing – lengths or coordinates.
Today, relativists are happy mentally to absorb c into t, and do calculations
in relativity theory with x, y, z, and t as four space-time coordinates
on an equal footing with each other. They do not need c.

Looked at from this point of view, we can argue that c is just a conversion
factor. To convert lengths in the fourth dimension into their equivalents
in the other three dimensions, we simply multiply them by a number. This
number is equal to the numerical value of c, but is not fundamental. What
we are used to thinking of as 5 seconds, for example, can be expressed equally
well as 5 ‘fourth-dimensional length units’. This happens to be equivalent
to 15 x 10 ⁸ metres. At school we were taught to think of this
length as the distance that light would travel in 5 seconds, at a speed
of 3 x 10 ⁸ metres per second. But we would argue that this 5
seconds is actually equivalent to a length of 15 x 10 ⁸ metres
and that the conversion factor c has no more significance than the one
we use to convert kilometres into miles (by multiplying the number of kilometres
by 0.62137119).

Could other constants be explained away in a similar fashion? What about
the constant of gravity, G, for example? It can, but things become a little
more complicated. As well as lengths and times, Newton’s law for the acceleration
a of a particle near a mass m involves the mass itself, and reads a = Gm/d
². We perceive mass as being different from space and time, and
quantify it in dimensions of (M). So with three different kinds of things
in the equation (acceleration involves length and time) we need a more
complicated constant. In terms of the base dimensions, G has to have dimensions
(L ³)/(M)(T ²). A constant with the same dimensions,
but including c, turns up in Einstein’s general theory of relativity, which
is a theory of gravity that contains Newton’s theory as a special case.

GETTING RID OF G

Can we get rid of G? The quantity Gm/c ² has the dimensions
of length, just as the quantity ct has the dimensions of length. If we absorb
G we can keep m, treating masses (as well as times) in terms of length.

With G and c out of the way, we have simplified relativity theory considerably.
And we can do the same for the other great theory of 20th-century physics,
quantum mechanics. The additional important fundamental constant in quantum
mechanics is Planck’s constant, h. This turns up most simply in the law
relating the energy of a wave packet (or photon) to its frequency: E = hv.
We also know from Einstein’s famous relation E = mc² that energy
is related to mass. Frequency, of course, is related to time (it is measured
in units of ‘per second’). Putting all the dimensions in, we find that h
has the form (M)(L ²)/(T). Just as c is made up of (L) and (T),
h is made up of (M), (L) and (T). It is possible to absorb mass and time
as we have done before, and just end up with a number, and so we can write
E = v.

We can also use another trick to get round the complications caused
by including fundamental constants. This goes back to Max Planck at the
beginning of the 20th century. He noticed that instead of using our familiar
systems of units (metres, seconds, kilograms) to define the values of the
constants, we can use h, c and G to define a set of ‘absolute’, or natural,
units.

Previously, we got rid of the dimensions of the constants by absorbing
them into fundamental quantities that cannot be subdivided, such as mass
or time, and by turning mass (and everything else) into length. But we kept
the numerical values of the constants as conversion factors. Planck (among
others) suggested removing the numerical importance of the constants, by
choosing units of measurements in which the value of each of the constants
is 1. This is done by combining h, c and G in different ways to produce
quantities we now call the ‘Planck length’, the ‘Planck time’ and the ‘Planck
mass’, which have the dimensions of pure length, pure time and pure mass.

For example, if we want to determine the natural length scale of the
physical world, the Planck length, we can form the combination (Gh/c ³)
^1/2, which has dimensions of length (L). The numerical value
of the Planck length is about 10 ^-35 metres, 20 orders of magnitude
smaller than the size of a proton. The equivalent Planck time is about 10
^-43 seconds, and the Planck mass is about 10 ^-8 kilograms.
(The gravitational radius of a black hole with the Planck mass would be
the Planck length.)

In these units, h, c and G all have the numerical value 1. For example,
the speed of light is 1 because it takes light 1 Planck time to travel across
1 Planck length. And there is no reason why we cannot also apply our previous
trick of converting everything into lengths. This isn’t just an abstract
supposition; relativists, in particular, very often work with equations
in which c, G, h and the rest are simply left out of the calculation. All
these ‘fundamental constants’ do is clutter up the equations, affecting
the way the calculations are carried through. Any physical quantity should
have both a meaningful size and meaningful dimensions. But c, G, and h do
not necessarily have either, and we could argue that physicists can do without
these constants if they choose.

All this sounds a bit esoteric, of interest only to relativists and
quantum theorists. But there is a simpler example. It concerns a constant
known as the permittivity of free space, epilson, which is a measure of
how well empty space transmits an electric field. Over the years this quantity
has appeared and disappeared from electromagnetism, according to the prevailing
convention in units and how tidy physicists want their equations to look.
At present, SI units are de rigueur. In these units, Coulomb’s law for the
force between two electric charges, q₁ and q₂, separated by a distance d,
reads F = q₁q₂/4&pgr;&egr;d². But &egr; only appears here
if we regard charge as a different concept from mass, length and time. If
we wish, we can use cgs units (based on the centimetre, gram and second),
or old-fashioned electrostatic units, in which case Coulomb’s law reads
F = q₁q₂/d².

Where has &egr; gone? It has been absorbed into the charge, which,
in these units, has dimensions (M ^1/2)(L^3/2)/(T);
there is no distinct dimension ‘charge’. &egr; is merely a conversion
factor related to a choice of units, and we can do calculations involving
electromagnetism perfectly well without it.

Having discovered that electromagnetism does not need &egr;, it may
not come as too much of a shock to learn that gravitational theory does
not need G. In 1986 William McCrea, in his Milne Lecture (Quarterly Journal
of the Royal Astronomical Society, vol 27, p 137) forcefully made the point
that G only appears in our equations if we choose to treat mass in a different
way from length. Like c, it is a conversion constant, and nothing more.

But if everything can be expressed in terms of length, what (and where)
are these lengths? You can pick up a pen, say, and feel its weight as something
different from the length which is seen by your eyes. The mass of the pen
is not simply an ‘extra’ length in our familiar three-dimensional world.
So, just as we need a fourth dimension for time, perhaps we need extra dimensions
of space for these extra lengths which are invisible to our eyes. Each dimension
would correspond to a physical quantity such as mass. This is not a new
idea, but it is one which has undergone something of a revival recently.
It offers several different lines of attack on the problem of finding a
satisfactory description of the physical world.

NEW DIMENSIONS

This approach is often called the Kaluza-Klein theory, after the two
pioneers who in the 1920s first extended Einstein’s four-dimensional theory
of gravity (general relativity) into higher dimensions. (Here, ‘dimensions’
has its familiar sense of directions at right angles to each other – the
three directions, or dimensions, of space, one of time, and so on. This
is not the same as ‘dimension’ as used to describe the physical qualities
of a property such as mass or length.)

Einstein’s general theory of relativity explained what we perceive as
the force of gravity in terms of the curvature of four-dimensional space-time
(¿ìè¶ÌÊÓƵ, Inside Science 31, 24 February 1990). The initial triumph
of the Kaluza-Klein theory in the early 1920s was that it explained electromagnetism
by applying exactly the same equations as those of Einstein’s general theory,
but in a five-dimensional context; it explained electromagnetism in terms
of five-dimensional space-time.

The idea caused some excitement at the time, when the only known physical
forces were gravity and electromagnetism. But the discovery in the 1930s
of the strong and weak forces that only operate within atomic nuclei made
the Kaluza-Klein theory seem less attractive. Adding these forces to the
equations requires the addition of not just two but at least half a dozen
‘new’ dimensions to the equations. Such multidimensionality emerges naturally,
however, in some modern cosmological theories (superstring theory, for example)
and this has encouraged a new look at the Kaluza-Klein idea.

In the first paper on this work, in 1921, Theodor Kaluza showed that
classical electromagnetism can be unified with general relativity in a five-dimensional
world. In 1926, Oskar Klein extended this work to include quantum influences.
By considering the equations of motion of a particle with charge q, he
realised that the quantity ((c²/G)^1/2q) has the
dimensions (M)(L)/(T). If, as described earlier, we absorb the constants
c and G, we can regard the charge of a particle as a momentum. This momentum
‘belongs’ in the fifth dimension; but where is this dimension? The common
assumption mathematicians make in this kind of work, which Klein first suggested
in the mid-1920s, is that the extra dimension has been rolled up, or ‘compactified’,
to become invisibly small.

A good analogy is a hose pipe. Close up, a hose pipe is made of a two-dimensional
sheet wrapped into a tube around the third dimension. But from a great distance,
the hose pipe looks like a one-dimensional line. In the same way, a five-dimensional
entity may look like a four-dimensional or three-dimensional entity if it
is wrapped tightly around itself and viewed from a large distance compared
with the radius it is curved around. The relevant radius in Kaluza-Klein
theory may be the Planck length, 10 ^-35 metres – too small to
be visible to our senses and our scientific experiments (see ‘The fifth
dimension of mass’, ¿ìè¶ÌÊÓƵ, 22 September 1988).

Not all fundamental constants have dimensions attached. Indeed, the
number usually regarded as the prime parameter in particle physics, &agr;, is
dimensionless. This is the fine structure constant (&agr; = &egr; ²/ħ³¦,
where ħ = h/2&pgr;). In spectroscopy, ‘fine structure’ is the splitting of
spectral lines into close pairs, due to the fact that an electron can have
either of two spin states. The fine structure constant can be calculated
from measurements of this splitting and is related to the strength of the
interaction between electrons and photons. But this ‘fundamental constant’
has a curious property: it is not constant! This may sound odd: it is the
similarity between details of the spectra of light from atoms in distant
galaxies and those from identical atoms here on Earth that encourages physicists
to believe that the laws of physics are the same everywhere. But this physics
still leaves scope for &agr; to vary.

The constant alpha describes the strength of the interaction between
two electrons (we can see this from the e ² in the equation).
If the superfluous parameters h and c are absorbed as before, a becomes
a measure of the basic charge e. But experiments show that the effective
charge of an electron is stronger the closer you get to it – in other words,
a is bigger for electrons that are closer together.

NOT QUITE A CONSTANT

How can we understand this? In quantum physics, ’empty space’ is a place
where pairs of particles of opposite charge are constantly being created
and destroyed. This is allowed by Heisenberg’s uncertainty principle, provided
such ‘virtual’ particle pairs wink in and out of existence so fast that
the Universe does not ‘notice’ them. Even so, there is still time for the
positively charged partner in a virtual pair to edge closer to a nearby
electron, and for the negatively charged partner to be marginally repelled.
So a negatively charged particle such as an electron is envisaged as surrounded
by electric dipoles whose negative ends are further out from the particle
than their positive ends. This screens the effective charge of the particle,
so that it seems less at greater distances than at smaller ones; in other
words,&agr; is larger closer in, so it is not a constant.

We can apply equivalent arguments to other properties of particles
in the quantum world. Dimensionless coupling ‘constants’ that in fact vary
with distance appear in the highly successful modern theory of electromagnetism,
quantum electrodynamics (QED); an equivalent theory for the weak interaction,
responsible for radioactive decay; and in the colour interaction (quantum
chromodynamics, or QCD) which describes the forces between quarks, and
is responsible for the structure of the proton and neutron and the strong
nuclear force.

However, a puzzle is why the long-range value of e the same for all
electrons (and protons). How do they ‘know’ that this should be so, and
what the value should be? This remains one of the fundamental puzzles in
science. This is especially troublesome since, according to the simplest
cosmological models, parts of the Universe that are now in causal contact
with one another (that is, able to exchange light signals) used not to be
in communication. And yet, astrophysical information from quasars, and the
fact that the temperature of the 3 K microwave background is the same from
all parts of the sky, show that electrons that have never been in touch
with each other since the Universe was born all have the same e.

A partial answer to this puzzle may be in terms of ‘inflation’, the
idea that the Universe underwent a period of rapid expansion when young.
This allows light signals to travel farther than in standard models so more
electrons would ‘know’ about each other. But there are problems with this.
While it may explain the temperature of the 3 K microwave background, it
does not really explain why all electrons have the same value of e.

Physicists have come up with a few extravagant possibilities. Forty
years ago, Richard Feynman and John Wheeler mused that there might really
only be one electron in the Universe, capable of travelling through ‘wormholes’
in four-dimensional space-time and popping up almost everywhere at the same
time. Another idea is that we move about in three-dimensional space looking
through ‘holes’ into another dimension at a single electron at the centre
of the Universe.

The pioneering Cambridge physicist Arthur Eddington suggested more than
50 years ago that a lot of what passes for objective science is actually
subjective and the result of how we measure things. He used the analogy
of a fisherman who infers that there are no fish smaller than a certain
size in the sea, not realising that the size limit is actually set by the
size of the mesh in the net he is using. Perhaps it is time to change our
net for one with a finer mesh. Fundamental constants have proved convenient
historically, but they are all invented and it is possible to argue safely
that they should be removed from our equations. As Shakespeare wrote in
The Tempest:

These our actors, As I foretold you, were all spirits, and Are melted
into air, into thin air.

Paul Wesson is a physicist based at the University of Waterloo, in Ontario,
and also spends time at the University of California at Berkeley.

Further Reading Paul Davies and John Gribbin, The Matter Myth (Viking,
1992). The equations on which this article is based can be found in Paul
Wesson’s article in Space Science Reviews (vol 59, p 365, 1992).